Stirling permutations, marked permutations and Stirling derangements

TL;DR

This paper introduces marked permutations, establishes a bijection with Stirling permutations, and explores their properties through involutions and enumerative polynomials, providing new combinatorial insights.

Contribution

It defines marked permutations, links them to Stirling permutations via a bijection, and analyzes their enumeration and structure with new involutions and polynomial representations.

Findings

Established a bijection between Stirling and marked permutations

Presented an involution on Stirling derangements

Derived symmetric bivariate enumerative polynomials

Abstract

In this paper we introduce the definition of marked permutations. We first present a bijection between Stirling permutations and marked permutations. We then present an involution on Stirling derangements. Furthermore, we present a symmetric bivariate enumerative polynomials on $r$-colored marked permutations. Finally, we give an explanation of $r$-colored marked permutations by using the language of combinatorial objects.